Isaac Newton and the Laws of Motion and Gravitation 2 ASTR 101 3/21/2018 • Center of Mass motion • Oblate shape of planets due to rotation • Tidal forces and tidal locking • Discovery of Neptune 1 Center of Mass • It is often said that the Moon goes around the Earth or the Earth revolve around the Sun. • But that is not exactly correct. The gravitational attraction between two objects is mutual. • Just as the Moon feels the gravitational pull from the Earth, Earth is subjected to the gravitational pull from the The Moon goes around the Earth Moon. is an over simplified concept – So no reason the Earth stays in the middle and the Moon goes around it. • What happens is that both Earth and the Moon go around their common center of mass. • Center of mass is the average (or mean) location of all the mass in the system (single object or system made of The Earth and the Moon revolve few objects like the Earth and the Moon.) around their mutual center of mass • It is the point where an object would stay balanced. CoM CoM 2 Moon 384000 km 4600 km CoM + Earth 1 82 Moon CoM Earth • Center of mass is closer to the more massive object • For example: the Earth is 82 time more massive than the Moon. (M☾ = 22 kg, 24 7.35x10 M⊕ = 6x10 kg). • so the center of mass of the Earth-Moon system is 82 times closer to the Earth than the moon. • Center of mass of the Earth-Moon system is located within the Earth. To a good approximation it is like Moon moves around the Earth. • The Sun is over 700 times more massive than the masses of all planets combined. Therefore the center of mass of the solar system is very close to the Sun. – For most purposes we can consider planets are moving around the Sun. 3 m2 : m1 Images taken by the New Horizons spacecraft in shows the dwarf planet Pluto and its largest m1 + + m CoM d 2 moon, Charon, orbiting their common CoM. (orbital period ~6.4 days, images taken in January 2015, closest approach in July. ) www.nasa.gov/content/the-view-from-new-horizons-a-full-day-on-pluto-charon • There are binary stars and planetary systems with companions having similar masses. • When companions have comparable masses their center of mass is in the middle. 4 Oblate shape of the Earth • In Principia Newton suggested that the Earth could not be perfectly round: – due to inertial forces resulting from rotation, the Earth had to bulge at the equator: Shape of an oblate spheroid Without rotation gravity pulls matter rotation creates an inertial outward towards the center from all directions force. It is largest near the equator • Due to rotation, matter in the Earth moves in circular paths. Rotational velocity gets larger as the distance from the axis of rotation increases. • As a result, inertia of matter tends to pull them outward. (just like a ball pull the string when it is in circular motion). • It is zero at the poles and increases to a maximum at the equator. • Resulting shape of this inertial “centrifugal force” and the Earth’s gravity is an oblate spheroid. 5 • All rotating objects show such flattened shape. 12713 km 12756 km • Earth: • Polar diameter: 12713 km • Equatorial diameter: 12756 km, flattening 0.3% • Jupiter, and Saturn show much higher flattening due to their larger size and faster rotation. • Jupiter: flatting 6% (diameter 11d⊕, rotational period ~10 hours) • Saturn: flattening 10% (diameter 9.5 d⊕ rotational period ~10.6 hours) 6 Tidal Forces stronger weaker gravity gravity here here Gravitational pull relative to pull at the center, which is like a ‘differential force’ acting on it Which tries to stretch and squash it • Gravitational attraction between objects decrease with inverse square of the distance between them (1 푟2). • As a result when a large body (like Earth) is attracted to another (say Sun) it will experience a larger attractive force on the nearest side to the other than it will on the far side. • This difference in the gravitational force across an object creates stress in the object called Tidal forces, which tries to stretch the object. • Tidal forces could deform and tear apart objects 7 Roche Limit particles are at different distances are attracted differently by the planet at the same time are attracted toward planet each other due to their mutual gravity • The planet tries to tug the particles apart, while the particles themselves are trying to pull themselves together due to their own gravity. • The distance to the particles from the planet determines which process dominates. – Far from the planet mutual gravity of the particles can pull them together, particles coalesce to form a larger object (a moon) under their own gravity – Close to the planet, the planet's tidal forces trying to rip them apart are stronger than the mutual gravitational forces trying to keep them together, and tear them apart. • The borderline distance which separates the regions where an object (e.g. a moon) held together by its self-gravity can exist as a single object from the region where they get torn apart due to tidal forces is called the Roche Limit. Roche Limit Roche limit Particles coalesce to form a larger object under mutual gravity planet Tidal forces move apart particles and form a ring Roche limit Outside the limit material tends to coalesce, or stay together. , whereas Inside the Roche limit, orbiting material disperses and forms rings (like Saturn's rings) (Roche limits: Earth 18,470 km, Jupiter 175,000km, Saturn 147,000km) Tidal Forces apod.nasa.gov/apod/ap040724.html Comet Shoemaker-Levy torn into fragments by tidal forces before it collided with Jupiter. (Hubble space telescope photo taken on May 17, 1994) moon Io impact sites a collision Mars’ moon Phobos is orbiting just 6,000 km above the surface of Mars. Grooves visible on Phobos could be an indication of early signs of the structural failure produced by tidal forces. https://www.nasa.gov/feature/goddard/phobos-is-falling-apart 11 Roche limit Epimetheus Enceladus Mimas Rings: tidal forces Janus prevented forming moons Roche limit Saturn's rings and Moons Cassini spacecraft image of Saturn rings Rings are made of small particles (an illustration) apod.nasa.gov/apod/ap040723.html Saturn rings are made of small particles (from micrometers to meters in size), mostly ice particles. Since they are within the Roche limit, tidal forces prevent them coming together and forming large objects. Ocean Tides • Ocean tides are the phenomena of periodic rise and fall of the sea level. There are two high tides and two low tides every 24 h 50m. • They are a result of the tidal forces of the Moon and the Sun on Earth. • In Principia Newton explained what causing the tides. 13 sun’s pull on solid earth sun’s pull on ocean is is stronger than on the stronger than on the ocean so the earth is D solid Earth so water pulled away from pulled towards Sun ocean B C A Sun E Consider the effect of Sun’s gravitation on the Earth and the ocean: • The gravitational attraction to the Sun is stronger on the side closer to the sun (A). • The gravitational attraction to the Sun is weaker on the side opposite to the sun (B). – The difference in gravitational attraction tries to pull Earth apart. – Solid earth cannot move, but ocean water can flow, thus rising ocean levels both toward and away from the Sun (high tides). – When water pileup under and opposite sides of the sun, water levels drop (low tide) in other areas (D,E). 14 11770 d⊕ 30 d⊕ 8 Sun 1.5x10 km Earth 3.85x105km Moon d⊕=12740 km Similar mechanism work with the Moon’s gravitational pull on Earth: • Moon’s gravitational pull on Earth is weaker than that of the Sun. • But the Moon is closer to the Earth than the Sun. ⇒ difference in the Moon’s gravitational attraction on opposite sides of the Earth is larger ⇒ Twice stronger tidal forces. • Hence lunar tides are two times larger than solar tides. – High tides are experienced when the Moon is at the zenith or at the nadir. low tides – Low tides occur when the moon is at the horizon (east or west) Earth moon • Length of the lunar day is 24h 50m (That is the time for moon to come to same B C A position in the sky again.) – High tides due to the Moon occur at a given location every 12h 25m. Same with low tides. high tides solar tide + lunar tide or Spring tides Overall tide is a combination of solar and lunar tides : • When the Sun and the Moon line up combined lunar and solar tides reinforce each other causing largest tides called Spring tides – Happens on a full moon or new moon day 16 + solar tide lunar tide or Neap tides • When the Moon is at quarter phase, combined lunar and solar tides oppose each other, causing smaller tides called Neap Tides – Happens on a first quarter or last quarter moon 17 Tidal friction/acceleration in the Earth-Moon system rotation of earth Moon pull from Moon’s gravity If there is no friction between ocean and floor, tidal bulges are aligned in the Earth-Moon direction. • Since tidal force by the Moon is stronger we expect the tidal bulge always point directly towards the moon. – But due to the rotation of the Earth and the friction between ocean floor and water, lunar tidal bulge sits slightly ahead of the Earth-Moon line.